With regard to an internal combustion engine mounted as a running power source for a vehicle, a stop-start control apparatus is widely known which stops an internal combustion engine when a stop condition such as when a vehicle stops is fulfilled, and restarts it from its stopped condition when a restart condition is fulfilled in order to reduce amount of fuel consumption and amount of exhaust gas or the like in an idling operation. For example, as such a type of control apparatus, one restarts an internal combustion engine by making fuel injected into a cylinder in expansion stroke in a case that a restart request is made during an automatic stop operation period, thereby to making fuel be ignited and combusted, and further making fuel injected into a cylinder in compression stroke, thereby to making fuel ignited and combusted (Patent document 1). Furthermore, another restarts an internal combustion engine by making fuel filled in a cylinder in a stop process of an internal combustion engine and making the fuel ignited and combusted after the engine comes to a complete stop (Patent document 2).    Patent document 1: JP 2005-155362-A    Patent document 2: JP 2005-30237-A